Question

1. a 50-kg crate is attached to an electric hoist in a warehouse. the hoist begins lifting the crate with an average upward velocity of 4.0 m/s and maintains a constant upward acceleration of $1\\ \text{m/s}^2$. approximately how long will it take for the hoist to lift the crate to a shelf 10.0 meters above its starting position?

mass = 50 kg
velocity = 4.0 m/s
acceleration = $1\\ \text{m/s}^2$
$f_{crane}$
$f_g$
$f_{net}=f_{crane}-$

Explanation:

Step1: Select kinematic equation

We use the displacement equation for constant acceleration:
$$\Delta x = v_0 t + \frac{1}{2} a t^2$$
Where $\Delta x = 10.0\ \text{m}$, $v_0 = 4.0\ \text{m/s}$, $a = 1\ \text{m/s}^2$.

Step2: Substitute values into equation

$$10 = 4t + \frac{1}{2}(1)t^2$$
Rearrange to standard quadratic form:
$$\frac{1}{2}t^2 + 4t - 10 = 0$$
Multiply through by 2 to simplify:
$$t^2 + 8t - 20 = 0$$

Step3: Solve quadratic equation

Use the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=1$, $b=8$, $c=-20$:
$$t = \frac{-8 \pm \sqrt{8^2 - 4(1)(-20)}}{2(1)}$$
Calculate discriminant:
$$\sqrt{64 + 80} = \sqrt{144} = 12$$
Take positive root (time cannot be negative):
$$t = \frac{-8 + 12}{2} = \frac{4}{2} = 2$$

Answer:

$\boldsymbol{2\ \text{seconds}}$